Eliminating complex eigenvectors and eigenvalues in multiway analyses using the direct trilinear decomposition method

The direct trilinear decomposition method (DTDM) is an algorithm for performing quantitative curve resolution of three‐dimensional data that follow the so‐called trilinear model, e.g. chromatography–spectroscopy or emission‐excitation fluorescence. Under certain coditions complex eigenvalues and eigenvectors emerge when the generalized eigenproblem is solved in DTDM. Previous publications never treated those cases. In this paper we show how similarity transformations can be used to eliminate the imaginary part of the complex eigenvalues and eigenvectors, thereby increasing the usefulness of DTDM in practical applications. The similarity transformation technique was first used by our laboratory to solve the similar problem in the generalized rank annihilation method (GRAM). Because unique elution profiles and spectra can be derived by using data matrices from three or more samples simultaneously, DTDM with similarity transformations is more efficient than GRAM in the case where there are many samples to be investigated.